The Shape of Digits
A Bayesian Topological Data Analytic Approach to Classification of Handwritten Digits
Thomas Reinke
Theophilus A. Bediako
Daniel Lim
August 13, 2025
Contents
Exploratory Data Analysis
Distribution of training labels
Pixel Intensity
Training Data tSNE Visualization
Traditional ML
Neural networks
Feedforward neural network with structure:
- Input layer: Consists of neurons that receives the input data each neuron in the input layer represents a feature of the input data
- Hidden layer: One or more hidden layers placed between the input and output layers, responsible for capturing complex patterns
- Output layer: Final output of the network; Number of neurons represents the number of digits
NN with regularization
- Depending on model, # network weights > size of training data
- This leads to overfitting
- We considered two approaches to overfitting:
- Dropout learning: Like RF, randomly removes fraction of units in layer during model fitting
- Regularization: Impose penalties on parameters like lasso, ridge, etc.
Specific NN models considered
- NN with dropout regularization
- NN with ridge regularization
- NN with lasso regularization
Multinomial logistic regression
- Multinomial logistic regression equivalently represented by NN with no hidden layers
- Output layer with softmax
- \(f_m(X) = Pr(Y = m | X) = \frac{e^{Z_m}}{\sum_\limits{l \in K}e^{Z_l}}\)
NN Fitting
- Train the network for 30 epochs with a default batch size of 32
- SGD updates weights for each batch
- Images are presented in batches of 32, and SGD updates weights after each batch
- Each epoch processes all 60,000 training images
- Classification correct if largest output value matches target label
Proposed Methodology
TDA Workflow
TDA + ML
Analysis
ML Analysis
Proposed Method Analysis
TDA + ML Analysis
Results/Future Work
ML Results
| method | accuracy |
|---|---|
| multinomial | 0.9856 |
| dropout nn | 0.9962 |
| ridge nn | 0.9946 |
| lasso no | 0.9946 |
Propsed Results
ML + TDA Results
References
References
